Function G G G Is Represented By The Equation: G ( X ) = 4 ( 1 4 ) X + 2 G(x) = 4\left(\frac{1}{4}\right)^x + 2 G ( X ) = 4 ( 4 1 ​ ) X + 2 Which Statement Correctly Compares The Two Functions?A. They Have Different Y Y Y -intercepts But The Same End Behavior. B. They Have Different

Comparing Functions: A Deep Dive into $g(x) = 4\left(\frac{1}{4}\right)^x + 2$

When it comes to comparing functions, it's essential to understand the characteristics that define them. In this article, we'll delve into the world of functions, specifically the equation $g(x) = 4\left(\frac{1}{4}\right)^x + 2$. We'll explore the properties of this function, including its $y$-intercept and end behavior, and compare it to other functions to determine which statement correctly describes their relationship.

The function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ is a combination of two components: an exponential function and a constant. The exponential function $\left(\frac{1}{4}\right)^x$ represents a decreasing function, where the base $\frac{1}{4}$ is less than 1. This means that as $x$ increases, the value of the function decreases.

The $y$-Intercept

The $y$-intercept of a function is the point where the function crosses the $y$-axis, i.e., when $x = 0$. To find the $y$-intercept of $g(x)$, we substitute $x = 0$ into the equation:

$$g(0) = 4\left(\frac{1}{4}\right)^0 + 2$$

Since $\left(\frac{1}{4}\right)^0 = 1$, we have:

$$g(0) = 4(1) + 2 = 6$$

Therefore, the $y$-intercept of $g(x)$ is 6.

End Behavior

The end behavior of a function describes what happens to the function as $x$ approaches positive or negative infinity. For the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$, we can analyze its end behavior by considering the behavior of the exponential function $\left(\frac{1}{4}\right)^x$.

As $x$ approaches positive infinity, the value of $\left(\frac{1}{4}\right)^x$ approaches 0. This means that the function $g(x)$ approaches 2 as $x$ approaches positive infinity.

On the other hand, as $x$ approaches negative infinity, the value of $\left(\frac{1}{4}\right)^x$ approaches infinity. This means that the function $g(x)$ approaches infinity as $x$ approaches negative infinity.

Now that we've analyzed the properties of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$, let's compare it to other functions to determine which statement correctly describes their relationship.

Statement A: They have different $y$-intercepts but the same end behavior.

This statement is incorrect because the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a $y$-intercept of 6, but its end behavior is different from that of other functions with the same $y$-intercept.

Statement B: They have different $y$-intercepts and end behavior.

This statement is correct because the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a unique $y$-intercept of 6 and a distinct end behavior that approaches 2 as $x$ approaches positive infinity and infinity as $x$ approaches negative infinity.

In conclusion, the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a unique $y$-intercept of 6 and a distinct end behavior that approaches 2 as $x$ approaches positive infinity and infinity as $x$ approaches negative infinity. Therefore, the correct statement that compares the two functions is:

They have different $y$-intercepts and end behavior.

• The function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a $y$-intercept of 6.
• The function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a distinct end behavior that approaches 2 as $x$ approaches positive infinity and infinity as $x$ approaches negative infinity.
• The correct statement that compares the two functions is: They have different $y$-intercepts and end behavior.

• [1] Algebra, 2nd ed. by Michael Artin. Prentice Hall, 2010.
• [2] Calculus, 3rd ed. by Michael Spivak. Publish or Perish, 2008.

In our previous article, we explored the properties of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$. We analyzed its $y$-intercept and end behavior, and compared it to other functions to determine which statement correctly describes their relationship. In this article, we'll answer some frequently asked questions about the function $g(x)$ and its properties.

Q: What is the $y$-intercept of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$?

A: The $y$-intercept of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ is 6. This is because when $x = 0$, the function evaluates to $g(0) = 4\left(\frac{1}{4}\right)^0 + 2 = 4(1) + 2 = 6$.

Q: What is the end behavior of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$?

A: The end behavior of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ is that it approaches 2 as $x$ approaches positive infinity and infinity as $x$ approaches negative infinity. This is because the exponential function $\left(\frac{1}{4}\right)^x$ approaches 0 as $x$ approaches positive infinity, and approaches infinity as $x$ approaches negative infinity.

Q: How does the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ compare to other functions?

A: The function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a unique $y$-intercept of 6 and a distinct end behavior that approaches 2 as $x$ approaches positive infinity and infinity as $x$ approaches negative infinity. This makes it different from other functions with the same $y$-intercept.

Q: What is the domain of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$?

A: The domain of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ is all real numbers, i.e., $x \in (-\infty, \infty)$. This is because the function is defined for all real values of $x$.

Q: What is the range of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$?

A: The range of the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ is all real numbers greater than or equal to 2, i.e., $y \in [2, \infty)$. This is because the function approaches 2 as $x$ approaches positive infinity and approaches infinity as $x$ approaches negative infinity.

Q: Can the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ be used to model real-world phenomena?

A: Yes, the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ can be used to model real-world phenomena that exhibit exponential decay. For example, it can be used to model the decay of a radioactive substance or the growth of a population that is limited by a resource.

In conclusion, the function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a unique $y$-intercept of 6 and a distinct end behavior that approaches 2 as $x$ approaches positive infinity and infinity as $x$ approaches negative infinity. It can be used to model real-world phenomena that exhibit exponential decay. We hope that this Q&A article has provided a better understanding of the properties of the function $g(x)$ and its applications.

• The function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a $y$-intercept of 6.
• The function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ has a distinct end behavior that approaches 2 as $x$ approaches positive infinity and infinity as $x$ approaches negative infinity.
• The function $g(x) = 4\left(\frac{1}{4}\right)^x + 2$ can be used to model real-world phenomena that exhibit exponential decay.

• [1] Algebra, 2nd ed. by Michael Artin. Prentice Hall, 2010.
• [2] Calculus, 3rd ed. by Michael Spivak. Publish or Perish, 2008.

The author is a mathematics enthusiast with a passion for teaching and learning. They have a strong background in algebra and calculus and enjoy exploring the properties of functions.